Dinamical chaos in a nonlinear dielectric with the first
kind phase transition
Yu.N. Zayko 
, I.S. Nefedov 
Volga Region Academy of State Service, Saratov, Russia
Institute of Radio Engineering and Electronics of Russian
Ac. Sci., Saratov Branch
The idea of dynamical description of cooperative phenomena in particular phase transitions goes back to V.L. Ginzburg's works [1]. Here we develop this approach with the help of the Korteveg-de Vris-Burgers equation (KdVB) for the polarization of ferroelectric with the lattice free of symmetry center, for example KDP, which was derived in Ref. [2]:
where 
and 
are time and coordinate, respectively,
P is polarization, A,B and D are real coefficients.
The main feature of this KdVB equation is that some of its coefficients
have a pole at the 
, where 
is the Curie temperature.
This leads to the existence of an unlimited sequence of bifurcation
points where the n-th mode
of periodic solution grows abruptly for 
. This sequence has
a limiting point at , which allowed the analogy with the Feigenbaum
cascade [3]. With 
the coefficients of KdVB equation become
imaginary and Eq. (1) has the form (dimentionless):
Here we use a traveling wave type coordinate 
and new real
coefficients R and 
which govern the behavior of the system.
After substituting p=u+iw and a single integration we deduce from
Eq. (2):
where 
are the arbitrary constants.
Below we present the anlytical and numerical
investigation of the problem for T;SPMgt;Tc.
The main results of the investigation of Eqs. (3) are as follows:
1. The Eqs. (3) do not belong to the class of Hamiltonian systems,
i.e. it has no real Hamiltonian even for 
.
2. In spite of dissipation, Eqs. (3) conserve the phase volume unlike the typical dissipative systems.
3. There exists the curve 
which divides the plane of parameters
into two branches with different qualitative behavior
of Eqs.(3) solutions ( 
).
4. The behavior of the Kolmogorov-Sinay entropy lookes like the behavior
of thermodynamical entropy near .
5. Eqs.(3) (with 
) have no rigorous periodic solutions
or those which are assymptotically approaching them.
6. Despite the constant presence of chaotic component, solutions of Eqs.(3) can be classified similarly to the classification for a dynamical systems demontrating transition to chaos through the destruction of torus in phase space.
